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- W3118112987 abstract "Let μ be an Ahlfors-David probability measure on Rq with support Kμ. For every n≥1, let Cn(μ) denote the collection of all the n-optimal sets for μ with respect to the geometric mean error. We prove that, there exist constants d1,d2>0, such that for each n≥1, every αn∈Cn(μ) and an arbitrary Voronoi partition {Pa(αn)}a∈αn with respect to αn, we haved1n−1≤mina∈αnμ(Pa(αn))≤maxa∈αnμ(Pa(αn))≤d2n−1. Moreover, we prove that for each a∈αn, the set Pa(αn) contains a closed ball of radius d3|Pa(αn)∩Kμ| which is centered at a, where d3 is a constant and |B| denotes the diameter of a set B⊂Rq. Some estimates for the measure and the geometrical size of the elements of a Voronoi partition with respect to an n-optimal set are established in a more general context. These estimates also allow us to drop an additional condition in our previous work on self-similar measures." @default.
- W3118112987 created "2021-01-05" @default.
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- W3118112987 date "2021-06-01" @default.
- W3118112987 modified "2023-10-18" @default.
- W3118112987 title "On the optimal Voronoi partitions for Ahlfors-David measures with respect to the geometric mean error" @default.
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- W3118112987 doi "https://doi.org/10.1016/j.jmaa.2020.124897" @default.
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